Introduction
Mammalian mastication is a rhythmic, cyclic activity, the rhythmic components of which are controlled by at least one central pattern generator located in the lateral reticular formation in the brainstem or the principal sensory nucleus of the trigeminal (Lund et al., 1998; Tsuboi et al., 2003; Westberg et al., 2001). To facilitate effective occlusion without damaging the teeth, masticatory movements and forces must be adjusted in response to variation in material properties of the food, between and within chewing sequences and cycles (Agrawal et al., 2000; Ahlgren, 1976; Anderson et al., 2002; Blanksma and Van Eijden, 1995; Ottenhoff et al., 1993; Ottenhoff et al., 1992;
Ottenhoff et al., 1996; Peyron et al., 2002; Proschel and Hofmann, 1988). This active adjustment of movement and force by the central nervous system (CNS) in response to variation in some external variable is referred to here as modulation (Deban et al., 2001; Herrel et al., 2001). Evidence of modulation is usually derived from systematic covariation between external variables (e.g. sex of the subject, material properties of foods or objects placed between the teeth, external forces applied to the system) and the behavior of the subject, as quantified by kinematic or electromyographic variables (Anderson et al., 2002; Buschang et al., 2000; Hidaka et al., 1997; Lavigne et al., 1987; Lund et al., 1998; Morimoto et al., Modulation of force during mammalian mastication
provides insight into force modulation in rhythmic, cyclic behaviors. This study uses in vivo bone strain data from the mandibular corpus to test two hypotheses regarding bite force modulation during rhythmic mastication in mammals: (1) that bite force is modulated by varying the duration of force production, or (2) that bite force is modulated by varying the rate at which force is produced. The data sample consists of rosette strain data from 40 experiments on 11 species of mammals, including six primate genera and four nonprimate species: goats, pigs, horses and alpacas. Bivariate correlation and multiple regression methods are used to assess relationships between maximum (⑀1) and minimum (⑀2) principal strain magnitudes and the following variables: loading time and mean loading rate from 5% of peak to peak strain, unloading time and mean unloading rate from peak to 5% of peak strain, chew cycle duration, and chew duty factor. Bivariate correlations reveal that in the majority of experiments strain magnitudes are significantly (P<0.001) correlated with strain loading and unloading rates and not
with strain loading and unloading times. In those cases when strain magnitudes are also correlated with loading times, strain magnitudes are more highly correlated with loading rate than loading time. Multiple regression analyses reveal that variation in strain magnitude is best explained by variation in loading rate. Loading time and related temporal variables (such as overall chew cycle time and chew duty factor) do not explain significant amounts of additional variance. Few and only weak correlations were found between strain magnitude and chew cycle time and chew duty factor. These data suggest that bite force modulation during rhythmic mastication in mammals is mainly achieved by modulating the rate at which force is generated within a chew cycle, and less so by varying temporal parameters. Rate modulation rather than time modulation may allow rhythmic mastication to proceed at a relatively constant frequency, simplifying motor control computation.
Key words: bone strain, muscle recruitment, size principle, chewing.
Summary The Journal of Experimental Biology 210, 1046-1063
Published by The Company of Biologists 2007 doi:10.1242/jeb.02733
Modulation of mandibular loading and bite force in mammals during mastication
Callum F. Ross
1,*, Ruchi Dharia
2, Susan W. Herring
3, William L. Hylander
4, Zi-Jun Liu
3,
Katherine L. Rafferty
3, Matthew J. Ravosa
5and Susan H. Williams
61Organismal Biology and Anatomy, University of Chicago, 1027 E. 57th Street, Chicago, IL 60637, USA, 2Stony
Brook School of Medicine, Health Sciences Center Level 4, Stony Brook, NY 11794-8434, USA, 3Department of Orthodontics, School of Dentistry, University of Washington, Seattle, WA 98195-357446, USA, 4Department of Biological Anthropology and Anatomy, Duke University Lemur Center, Durham, NC 27710, USA, 5Department of
Pathology and Anatomical Sciences, University of Missouri School of Medicine, One Hospital Drive – Medical Sciences Building, Columbia, MO 65212, USA and 6Department of Biomedical Sciences, Ohio University College of
Osteopathic Medicine, 228 Irvine Hall, Athens, OH 45701, USA *Author for correspondence (e-mail: [email protected])
1989; Nakajima et al., 2001). These studies reveal that descending control and afferent information from muscle spindles, Golgi tendon organs, and stretch receptors in the periodontal and ‘periarticular’ ligaments modulate the motor program and temporal and spatial patterns of jaw movement (Sessle, 2006).
Modulation of the magnitude and orientation of the bite force is less well documented, primarily because bite force is difficult to measure directlyin vivoduring mastication. Some workers have related variation in EMG patterns to variation in bite force orientation and magnitude during isometric biting on a bite force transducer (Blanksma et al., 1997); others have measured bite force orientations and/or magnitudes from intra-oral implants in edentulous (Mericske-Stern et al., 1992) or partially dentate (Lundgren and Laurell, 1986) subjects. These studies have various limitations, the most serious of which is destruction of the periodontal ligament afferents employed to modulate bite force production (Trulsson, 2006).
An alternative method for investigating bite force modulation is to use bone strain from the mandibular corpus below the molar teeth as an indirect estimate of changes in bite force (Hylander, 1977; Hylander, 1986; Weijs and De Jong, 1977). This method has the advantage of being easy to apply to a wide range of animals without significantly impacting normal masticatory function. Empirical support for the relationship between bite force and mandibular bone strain comes from Hylander’s investigation of mandibular corpus bone strain data during isometric biting on a force transducer (Hylander, 1977; Hylander, 1979; Hylander, 1986). Hylander reports “that when bite-point position is held constant, there is a high positive correlation between the magnitude of peak bite force and peak mandibular bone strain during isometric biting in both macaques and galagos”, concluding that “bone-strain patterns along the working side of the mandible are a good indicator of bite-force patterns during the power stroke” [Hylander (Hylander, 1986), p. 149]. These data suggest that mandibular corpus bone strain data may provide insight into modulation of bite force during mastication in mammals (Weijs and De Jong, 1977, p. 647).
Hypotheses
The primary aim of this study is to determine how bite force is modulated during rhythmic mastication in mammals. Because patterns and mechanisms of force modulation also impact other temporal aspects of chewing cycle dynamics, the relationship between corpus bone strain magnitude – a proxy for bite force – and both chewing cycle duration and chewing ‘duty factor’ are also evaluated (where duty factor is the percentage of the chewing cycle over which the jaw is loaded). Specifically, the present study used bone strain data from a range of mammals to examine how strain magnitude in the mandibular corpus and, by inference, bite force, are related to the loading and unloading rates, loading and unloading times, overall chew cycle time, and chewing ‘duty factor’.
The possible relationships between strain magnitude, and loading time and rate are easily understood by modeling the loading portions of the strain profile as a triangle, with strain magnitude represented by the vertical limb of the triangle, loading time by the base, and loading rate as strain magnitude/loading time (Fig. 1).
Strain magnitude can be increased by increasing loading time while maintaining a constant load rate (top row, Fig.·1), increasing load rate while maintaining a constant load time (bottom row, Fig.·1), or some combination of the two. If strain magnitude is increased by increasing loading time (i.e. time-modulated), then either chewing duty factor must increase, overall cycle time must increase (i.e. chewing frequency must decrease), or both. This hypothesis predicts that variation in strain magnitude will be positively correlated with variation in load time, will not be correlated with strain rate, and will be correlated with increases in chewing duty factor and cycle time. Empirical support for this hypothesis derives from studies demonstrating that increases in hardness of objects placed between the teeth during cortically evoked rhythmic jaw movements (CRJMs) are associated with increases in cycle time (Hidaka et al., 1997; Lavigne et al., 1987; Liu et al., 1998; Liu et al., 1993). Time-modulation of force is also suggested by data showing that when harder foods are chewed there are increases in cycle time, increases in the duration of the slow close or power stroke phase of the chewing cycle, and/or increases in burst durations of jaw adductor muscles (Kakizaki
200 100 500
400 300 200 100
500
A
B
400
300
Str
ain magnitude
Time
et al., 2002; Weijs and Dantuma, 1981; Yamada and Haraguchi, 1995; Yamada and Yamamura, 1996).
If strain magnitude is increased by increasing load rate without increasing load time (i.e. rate-modulated), cycle time will remain relatively constant and there need not be changes in cycle time or duty factor. This hypothesis predicts that variation in strain magnitude will be positively correlated with variation in strain rate, and not with variation in load time, chewing duty factor, or cycle time. Theoretical support for this hypothesis derives from considerations of the consequences of orderly recruitment of motor units, the primary mechanism for force modulation at low force amplitudes during mastication (Goldberg and Derfler, 1977; Hannam and McMillan, 1994; Scutter and Türker, 1998) and locomotion (Fournier and Sieck, 1988; Hennig and Lomo, 1987; Tansey et al., 1996). If the small motor neurons, which innervate small motor units consisting of slow twitch fibers, are recruited first, followed by progressively larger motor neurons and motor units consisting of faster fiber types, then increases in bite force will necessarily be achieved in a constant or decreasing time period. The evidence for the orderly recruitment of motor units during mastication is summarized in the Discussion. Weijs and DeJongh have presented empirical support for this hypothesis (Weijs and DeJongh, 1977), and their data show that differences in strain magnitudes when chewing different foods are accounted for by differences in strain rate. In addition, their EMG data (Weijs and Dantuma, 1981) reveal that increases in vertical components of jaw elevator muscle force are achieved via increases in rate of force development.
While modulation of bite force and correlated changes in bone strain magnitudes during the loading portion of the chewing cycle is to be expected in theory and is suggested empirically, it is less clear that modulation during the unloading portion of the chewing cycle will be functionally related to bite force modulation. Muscles acting to move the teeth out of occlusion (e.g., balancing side deep masseter) and open the jaws (digastric, mylohyoid, geniohyoid) do strain the mandible after centric occlusion, and modulation of mandibular loading might be expected in relation to these forces. However, strain during unloading of the mandible is thought to be primarily affected by the relaxation characteristics of the jaw elevator muscles (Hylander and Johnson, 1993; Hylander and Johnson, 1994; Hylander et al., 1987; Luschei and Goodwin, 1974).
Materials and methods
Data sample
The mandibular bone strain data analyzed for this study (Table·1) were recorded in three different laboratories in connection with other studies (Herring and Teng, 2000; Hylander et al., 1998; Liu et al., 2004; Liu and Herring, 2000a; Liu and Herring, 2000b; Rafferty, 2005; Ravosa et al., 2000; Ross and Hylander, 1996; Thomason et al., 2001; Williams, 2004). Data were available for 3084 chewing cycles from 31 individuals of 11 species, including seven species of primates (four anthropoid and three strepsirrhine species), as well as
goats (Capra), horses (Equus caballus), alpacas (Lama pacos), and miniature pigs (Sus) (Table·1). The data were collected in 40 experimental sessions, referred to hereafter as ‘experiments’. Data were collected from four of the primate subjects in two separate experiments.
The data were collected with either delta or rectangular rosette strain gages. Gage location varied between experiments, although all the data discussed here were collected from the lateral aspect of the mandibular corpus below the molars or premolars. Some gages were placed close to the lower (ventral) border of the mandibular corpus, others at mid-corpus height (Table·1). The gages were placed on the mandibular corpora with the animals under sedation with ketamine (primates at Duke University); or anesthesia with isoflurane (primates at Stony Brook University); isoflurane and nitrous oxide (goats at University of Washington); 5·mg·kg–1ketamine, 0.05·mg·kg–1 butorphenol, 0.5·mg·kg–1 xylazine intramuscularly (IM) (alpacas at Duke University); medetomidine 0.03-0.05·mg·kg–1 IM reversed with atipamezole IM (goats at Duke University); or halothane and nitrous oxide (pigs at University of Washington). The horses at Duke University were tranquilized with 1.1·ml·kg–1xylazine intravenously (IV), anesthetized with 2.0·ml·kg–1 ketamine IV, and an additional 2.0·ml·kg–1 ketamine administered IV every 10–15·min, as needed (Riebold et al., 1995).
The periosteum was scraped away from the gage sites, the bone degreased, and the gages bonded to the bone using cyanoacrylate adhesive. The lead wires were run out through the incision sites, which were sutured closed. The gage elements were connected to standard strain amplifiers, i.e. as one arm of a Wheatstone bridge, and the elements calibrated using shunt calibrations. Various other kinds of data were often recorded simultaneously, such as electromyographic data, or strain data from other sites, but these data are not discussed here.
After the animals recovered from anesthesia, strain data were recorded the same day. The animals were presented with a range of food types, depending on the species (Table·1). Data were either recorded on analog tape and digitized later using A/D boards in personal computers (e.g. data from the Hylander laboratory at Duke University), or digitally recorded directly to computer (data from the Ross and Herring laboratories). The digitizing frequencies are given in Table·1. Using the calibration files, the raw data files were converted to microstrain. The rosette data were used to calculate magnitudes and orientations of principal strains. Strain (⑀), a dimensionless unit equaling the change in length of an object divided by its original length, is measured in microstrain (⑀) units that are equal to 1⫻10–6strain. Tensile strain is registered as a positive value, and compressive strain as a negative value. The maximum principal strain (⑀1) is usually the largest tensile strain value, while the minimum principal strain is usually the largest compressive strain value (⑀2).
(1) Peak strain magnitude: the magnitude of the largest values of ⑀1 and ⑀2during the closing stroke;
(2) Peak strain timing: the time at which peak strain magnitudes (⑀1 and ⑀2) were reached;
(3) 5% timing: the time at which 5% of peak strain magnitude was reached in loading and unloading;
(4) Load time: the amount of time between 5% of peak strain in loading and peak strain;
(5) Cycle time: the duration of the chewing cycle, estimated as the average of the times from the preceding peak to the current peak, and from the current peak to the following peak; (6) Unload time: the amount of time between peak strain and 5% of peak in unloading;
(7) Standardized load time (load time STD): load times expressed as a percentage of overall cycle time;
(8) Duty factor: total load time as a percentage of chewing cycle time, i.e. the percentage of the cycle during which the mandible is loaded;
(9) Load rate: for each chew cycle, the average loading rate between 5% of peak and peak, i.e. ␦y/␦x, where ␦y=peak magnitude and ␦x=load time
(10) Unload rate: for each chew cycle, the average unloading rate between peak strain and 5% of peak strain, i.e. ␦y/␦x, where ␦y=peak magnitude and ␦x=unload time
The duration of loading, unloading, loading rate and
unloading rate are measured from the first occurrence of 5% of peak, rather than zero, strain in each power stroke because the strain profile often does not drop to zero between the strain associated with the opening phase and that associated with closing. These strains are particularly prevalent in taxa that chew rapidly and experience significant corpus strains during jaw opening (e.g. Fig.·2).
Ingestion cycles (cycles in which food is brought into the mouth) were excluded and data were only included from chewing cycles in which the animals chewed ipsilateral to the corpus strain gages; i.e. only working side bone strain data are examined here. Chewing side was determined using EMG patterns and/or jaw kinematic data. Chewing sequences were only selected for study if there were five or more cycles of rhythmic chewing. Because cycle time is calculated using the relative timing of three successive strain peaks, data on strain magnitude, rate and duration from the first and last non-ingestion cycles in each chewing sequence were not analyzed, although these cycles were used to calculate cycle durations/ frequencies for neighboring cycles. Most of the data sets did not allow puncture-crushing cycles, tooth–tooth contact cycles and swallowing cycles to be distinguished, so the data set includes all three. Primates are known to intercalate multiple swallows into a chewing sequence (Hylander et al., 1987; Thexton and Hiiemae, 1997), as may other species as well.
Data from all food types were pooled and analyzed together. Analyses of the data separated by food type revealed no food-related deviations from the patterns reported below.
Statistical analyses
To determine whether bite force is modulated by changes in load time and/or load rate, and whether there are resulting changes in chewing duty factor and cycle time, bivariate correlation coefficients were calculated within each experiment between strain magnitude (⑀1 and ⑀2) and load time, load time STD, load rate and cycle time. To investigate relationships between strain magnitude and strain profiles in unloading, correlations between strain magnitude and unload time, and unload rate were also calculated. To determine whether bite force is modulated through changes in duty factor, correlations between strain magnitude and duty factor were also calculated. If the data did not meet the assumptions of parametric statistics (i.e. skewness and kurtosis), either when untransformed or when transformed to log10, Spearman’s rho was calculated. Correlations between normally distributed and homoscedastic data combinations were estimated using Pearson’s r. For each set of comparisons, significance was assessed relative to a critical value of P=0.001 [0.05/(n experiments)= 0.05/ 40= 0.001], as well as the standard, less conservative, levels of significance, P⬍0.05, P⬍0.01.
Multiple regression models were run with strain magnitude as the dependent variable and load time and strain rate as the independent variables to determine which of these variables has the greatest influence on strain magnitude. This was assessed using beta coefficients. Beta coefficients are standardized
300
250
200
150
100
50
0
Str
ain magnitude (
με
)
28.2 28.0
27.8 27.6
Time (s)
ε1 peak magnitude
Time of
5% of peak Time of peak
T1 T2 T3
Fig.·2. Illustration of variables extracted from the strain data. Plot of
⑀1magnitude recorded from lateral aspect of the mandibular corpus
Table · 1. S o u
rces of d
a t a u sed i n t h is st u d y Species Investi g ators ID 5
Experiment ID n
u mber 6 Strain g a g e type Ga g e location
Sample rate (Hz)
Foods L ama p a cos Williams 1 P erla 40 Delta rosette 10 Midcorp u s1 0 · 000 Hay Williams 1 Geneviese 50 Delta rosette 10 Below M 1 , 4–10 · mm
above lower border
10 · 000 Hay Williams 1 La u ra 52 Delta rosette 10 Below M 1 , 4–10 · mm
above lower border
10 · 000 Hay Williams 1 Dabreon 56 Delta rosette 10 Midcorp u s1 0 · 000 Hay Su s scrof a (Hanford miniat u re strain) Li u /Herrin g 2 Bess 101 Rectan gu lar rosette 8
Below the 1st molariform tooth
500 P i g chow Li u /Herrin g 2 Freyr 102 Rectan gu lar rosette 8
Below the 1st molariform tooth
500 P i g chow Li u /Herrin g 2 Lily 103 Rectan gu lar rosette 8
Below the 1st molariform tooth
500 P i g chow Li u /Herrin g 2 Loci 104 Rectan gu lar rosette 8
Below the 1st molariform tooth
500 P i g chow Li u /Herrin g 2 Rama 105 Rectan gu lar rosette 8
Below the 1st molariform tooth
500 P i g chow Li u /Herrin g 2 Thor 106 Rectan gu lar rosette 8
Below the 1st molariform tooth
500 P i g chow Eq uu s c a b a ll u s Williams 1 B u cky 35 Delta rosette 10 Below M 2 , 4 · mm above lower border 10 · 000 Hay Williams 1 Domino 53 Delta rosette 10 Below M 2 , 8 · mm above lower border 10 · 000 Hay C a pr a c a pr a ( g oat) Rafferty/Herrin g 2 2 12 Rectan gu lar rosette 8 250 Hay Rafferty/Herrin g 2 4 14 Rectan gu lar rosette 8 250 Sweet g rain Rafferty/Herrin g 2 3 16 Rectan gu lar rosette 8 250 P i g chow Williams 1 Domin g o 20 Delta rosette 10 Below M 2 , midcorp u s1 0 · 000 Hay Williams 1 Don Q u ijote 33 Delta rosette 10 Below M 3 or M 2 , 7–12 ·
mm above lower
border 10 · 000 Hay M a c a c a f a scic u l a ris (lon g Hylander 3 Alex (1) 79 Rectan gu lar rosette 9
Lower border, M
1 500 Apple tailed macaq u e) Hylander 3 Mr Bi g (4) 791 Rectan gu lar rosette 9
Lower border, M
1,2 500 Apple Hylander 3 Alex (1) 81 Delta rosette 10 500 Apple Hylander 3 Dole (2) 83 Delta rosette 10 500
Apple, monkey chow
Hylander 3 Ni g htlady (3) 84 Delta rosette 10 500
Apple, monkey chow
M a c a c a mu l a tt a (Rhes u s macaq u e) Ross 4 Edith 38 Delta rosette 10 Lower border, P4 2700
Almond, apricot, pr
u ne Ross 4 Laverne 46 Delta rosette 10
Lower border, M
1
2700
Apple, jawbreaker, taffy
Ross 4 Shirley 48 Delta rosette 10 5 ·
mm from lower border, M
1 2700 Apple, g rape Table contin u
ed on next pa
g
Table
·
1.
Co
n
ti
nu
ed
Species
Investi
g
ators
ID
5
Experiment ID n
u
mber
6
Strain
g
a
g
e type
Ga
g
e location
Sample rate (Hz)
Foods
C
h
loroceb
u
s
a
et
h
iops
Ross
4
Male
59
Delta rosette
10
Mid corp
u
s,
P4
1000
Apple,
g
rape, pr
u
ne
(vervet monkey)
Ross
4
Male
96
Delta rosette
10
Lower border, M
1
10
·
000
Apple corn, pineapple, pr
u
ne
Aot
u
s tri
v
ir
ga
t
u
s
(owl
monkey)
Ross/Hylander
3
1
9
Delta rosette
10
5
·
mm from lower border, M
1
10
·
000
Apple, apricot, carrot, gu
mmy bear,
plantains, pr
u
ne
Ross/Hylander
3
2
10
Delta rosette
10
Mid corp
u
s, M
1
10
·
000
Apple,
gu
mmy bear,
pr
u
ne
Ross
4
3
31
Delta rosette
10
Lower border, M
2
2700
Apple,
gu
mmy bear,
pr
u
ne
Otole
mu
r cr
a
ssic
au
d
a
t
u
s
Ravosa/Hylander
3
Elf
u
921
Delta rosette
10
Lower border, M
1-2
500
G
u
mmy bear
(
g
reater b
u
shbaby)
Ravosa/Hylander
3
Marjoni
922
Delta rosette
10
Mid corp
u
s, M
1
500
P
r
u
ne
Ross
4
SB female
28
Delta rosette
10
? M
3
600
G
u
mmy bear
Ross
4
Vand females
63
Delta rosette
10
Lower border, M
2
2700
G
u
mmy bear, raisin
E
u
le
mu
r f
u
l
vu
s r
u
f
u
s
(brown
Ross
4
Bao
67
Delta rosette
10
Mid corp
u
s, M
2
1000
Apple,
g
rape, raisin
ss
o
R
)r
u
me
l
4
Beby
70
Delta rosette
10
Two
g
a
g
es: Mid corp
u
s,
M
2
, Mid corp
u
s,
P4
1000
Apple,
g
rape, pr
u
ne,
raisin
Ross
4
Haja
71
Delta rosette
10
Mid corp
u
s, M
1
1000
Apple,
g
rape, raisin
V
a
reci
a
va
rie
ga
t
a
r
u
br
a
(red
Ross
4
Borealis
91
Delta rosette
10
Mid corp
u
s, M
1
1000
P
ineapple, raisin
ss
o
R
)r
u
me
l
de
ff
u
r
4
Diphda
93
Delta rosette
10
Lower border, M
1
1000
Apple, raisin
Ross
4
Borealis
94
Delta rosette
10
Lower border, M
1
1000
Strawberry, pr
u
ne,
raisin
M
1
, 1st lower molar; M
2
, 2nd lower molar; M
3
, 3rd lower molar;
P4
, 4th lower premolar.
1 D
u
ke University, Research Farm of the Department of Lab Animal Reso
u
rces located in D
u
rham, NC, USA.
2University of Washin
g
ton.
3 D
u
ke University, Biolo
g
ical Anthropolo
g
y and Anatomy.
4Stony Brook University. 5 ID is animal ID
g
iven by experimenter who collected data.
6Experiment ID is ID n
u
mber in Ross data files.
7 N11-FA-2-120-11, Showa. 8SK-06-030WR-120, Micro-Meas
u
rements, Ralei
g
h, NC, USA.
9 WA-06-030WR-120, Micro-Meas
u
rements.
10 SA-06-030WY-120, Micro-Meas
u
regression coefficients obtained when all variables are standardized by conversion to z-scores. Beta coefficients express the relative standardized strengths of the effects of the independent variables on strain magnitude. The multiple regression models were run on z-scores using the General Linear Model univariate procedure in SPSS 12.0 in order to test for interaction effects. These were not significant, so the data were run again using the Multiple Regression procedure in order to simultaneously estimate the best model without interaction effects, as well as to obtain diagnostics of multicollinearity. Multicollinearity between independent variables in a multiple regression equation (in this case, load rate and load time) has significant effects on estimates of their partial slope coefficients. Specifically, the correlation between the estimators of the partial slope coefficients is the inverse of the correlation between the independent variables (Berry and Feldman, 1985); consequently, a high degree of multicollinearity weakens conclusions regarding the relative impacts of the two independent variables on the dependent variable. To assess the degree of multicollinearity, the correlation coefficients between strain rate and load time are also presented, along with the ‘tolerance’ statistic calculated by SPSS. The tolerance for a variable is the proportion of the variance in that variable not accounted for by other independent variables in the model. A low value indicates that the variable contributes little to the model independent of the other variables, and is an indicator of multicollinearity between independent variables.
Various multiple regression iterations were run in order to find the model that best explains the variance in strain magnitude with all variables significant. Only data that met the assumptions of linear regression were analyzed.
Results
Bivariate correlations
The bivariate correlations between ⑀1 magnitude and load rate, load time, load time STD, cycle time, duty factor, unload rate and unload time are given in Table·2; the bivariate correlations between ⑀2 magnitude and load rate, load time, load time STD, cycle time, duty factor, unload rate and unload time are given in Table·3. The results of the analyses using raw and logged data are very similar, and only the results using raw data are presented here. Fig.·3, Fig.·4, Fig.·5, Fig.·6 illustrate the relationships between ⑀1 magnitudes and both load rate and load time in four experiments: Experiment 71 on Eulemur, Experiment 9 on Aotus, Experiment 103 on Sus, and Experiment 16 on Capra.
Strain magnitude is significantly correlated with strain rate in the majority of experiments. ⑀1 magnitude is significantly
(P⭐0.001) correlated with ⑀1 load rate in 32 out of 40
experiments, and ⑀2 magnitude is significantly (P⭐0.001) correlated with ⑀2 load rate in 31 out of 40 experiments. In contrast, ⑀1 magnitude is only significantly (P⭐0.001) correlated with ⑀1 load time, load time STD and cycle time in 9, 17, and 3 out of 40 experiments, respectively; and ⑀2
magnitude is only significantly (P⭐0.001) correlated with ⑀2 load time, load time STD, and cycle time in 8, 6, and 4 out of 40 experiments, respectively. ⑀1 magnitude was significantly (P⭐0.001) correlated with duty factor in 6 out of 40 experiments and ⑀2 magnitude was significantly (P⭐0.001) correlated with duty factor in 9 out of 40 experiments.
Strain magnitude is significantly correlated with unload rate in the majority of experiments (⑀1 magnitude, 34/40 experiments; ⑀2 magnitude, 32/40 experiments). ⑀1 magnitude was significantly (P⭐0.001) correlated with unload time in only 9 out of 40 experiments, and ⑀2 magnitude was
Fig.·3. Bivariate plots of ⑀1 magnitude in microstrain (⑀) against
loading time (in s) (A) and loading rate in ⑀·s–1(B). Data recorded
during Experiment 71 on Eulemur fulvuseating apple (black circles), grapes (grey circles) and raisins (open circles). Note that there is not a significant correlation between strain magnitude and loading time (A), but there is a significant correlation between strain magnitude and loading rate (B). Although these data are not presented here, it is clear that these patterns of relationship (or lack thereof) also apply within different food types, although the nature of the relationship (i.e. the slope) may vary across food types.
0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 Food Apple Grape Raisin
0 2000 4000 6000 8000
0 100 200 300 400 500 0 100 200 300 400 500
ε1
magnitude (
με
)
ε1 load rate (με s–1) ε1 load time (s)
A
Table · 2. Correl a tio n coefficie n ts for 1 d a t a Species Experiment Ma g nit u de v s rate Ma g nit u de v s load time Rate v s load time Ma g nit u de v s
load time STD
Ma g nit u de v s cycle time Ma g nit u de v s d u ty factor Ma g nit u de v s u nload time Ma g nit u de v s u nload rate Unload rate v s u nload time A ot u s 9 0.584 *** (243) 0.263 *** (243) –0.570 *** (243) 0.168 ** (243) 0.299 *** (384) 0.383 *** (167) 0.527 *** (155) 0.770 *** (155) –0.056 NS (155) Aot u s 10 0.767 *** (181) –0.248 *** (181) –0.790 *** (181) –0.314 *** (181) 0.089 NS (195) –0.117 NS (167) 0.432 *** (169) 0.600 *** (169) –0.380 *** (169) A ot u s 31 0.596 *** (113) 0.520 *** (113) –0.260 ** (113) 0.027 NS (113) 0.232 * (119) –0.051 NS (103) 0.284 ** (97) 0.470 *** (97) –0.618 *** (97) C a pr a 12 0.668 *** (24) –0.023 NS (24) –0.698 *** (24) –0.003 NS (24) –0.244 NS (24) –0.017 NS (20) –0.354 NS (23) 0.658 *** (23) –0.917 *** (23) C a pr a 14 0.897 *** (24) –0.481 * (24) –0.696 *** (24) –0.420 * (24) 0.073 NS (28) –0.633 *** (28) –0.088 NS (21) 0.924 *** (21) –0.452 * (21) C a pr a 16 0.906 *** (22) –0.368 NS (22) –0.681 *** (22) –0.542 ** (22) 0.540 ** (22) –0.684 ** (16) 0.434 * (22) 0.706 *** (22) –0.076 NS (22) C a pr a 20 0.745 *** (47) 0.232 NS (47) –0.434 ** (47) 0.274 NS (43) –0.058 NS (44) 0.408 ** (44) 0.320 * (47) 0.768 *** (47) –0.321 * (47) C a pr a 33 0.375 * (45) 0.582 *** (45) –0.414 ** (45) 0.554 *** (41) –0.349 * (41) –0.029 NS (26) 0.280 NS (45) 0.596 *** (45) –0.591 *** (45) C h loroceb u s 59 0.345 * (35) 0.379 * (35) –0.700 *** (35) 0.454 ** (35) 0.081 NS (63) 0.204 NS (14) –0.065 NS (18) 0.660 ** (18) –0.517 * (18) C h loroceb u s 96 0.571 *** (86) 0.036 NS (86) –0.735 *** (86) –0.084 NS (86) 0.108 NS (87) 0.109 NS (75) 0.530 *** (86) 0.675 *** (86) –0.183 NS (86) Eq uu s 35 0.906 *** (38) –0.548 *** (38) –0.841 *** (38) –0.413 * (34) 0.164 NS (36) 0.008 NS (33) 0.360 * (39) 0.630 *** (39) –0.469 ** (39) Eq uu s 53 0.548 *** (54) –0.080 NS (54) –0.874 *** (54) 0.262 NS (52) –0.380 ** (54) 0.213 NS (51) 0.233 NS (53) 0.113 NS (53) –0.916 *** (53) E u le mu r 67 0.784 *** (81) 0.181 NS (81) –0.456 *** (81) 0.186 NS (81) –0.322 ** (93) 0.270 * (60) –0.384 *** (92) 0.918 *** (92) –0.691 *** (92) E u le mu r 70 0.472 *** (75) 0.205 NS (75) –0.737 *** (75) –0.036 NS (75) 0.251 * (83) –0.251 * (75) 0.284 * (73) 0.743 *** (73) –0.411 *** (73) E u le mu r 71 0.701 *** (233) –0.080 NS (233) –0.719 *** (233) –0.115 NS (233) 0.061 NS (235) –0.138 * (230) –0.128 NS (220) 0.783 *** (220) –0.643 *** (220) L ama 40 0.522 ** (27) 0.071 NS (27) –0.761 *** (27) –0.281 NS (25) 0.328 NS (25) –0.235 NS (25) 0.173 NS (27) 0.367 NS (27) –0.839 *** (27) L ama 50 0.917 *** (65) –0.500 *** (65) –0.740 *** (65) –0.462 *** (58) –0.148 NS (62) 0.452 ** (49) 0.196 NS (64) 0.937 *** (64) –0.116 NS (64) L ama 52 0.534 ** (32) 0.386 * (32) –0.492 ** (32) 0.271 NS (30) 0.245 NS (30) 0.102 NS (29) 0.047 NS (31) 0.581 *** (31) –0.666 *** (31) L ama 56 0.795 *** (17) –0.012 NS (17) –0.598 * (17) –0.259 NS (15) 0.423 NS 1 1 2. 0 ) 5 1( NS (16) 0.859 *** (16) –0.309 NS (16) M a c a c a 38 –0.096 NS (152) 0.576 *** (152) –0.832 *** (152) 0.643 *** (152) –0.292 *** (167) 0.427 *** (158) 0.261 *** (159) 0.301 *** (159) –0.801 *** (159) M a c a c a 46 0.519 *** (59) 0.108 NS (59) –0.745 *** (59) 0.373 ** (59) –0.104 NS (62) 0.471 *** (54) 0.029 NS (56) 0.741 *** (56) –0.577 *** (56) M a c a c a 48 0.296 ** (80) 0.366 *** (80) –0.738 *** (80) 0.269 * (80) 0.213 * (95) –0.006 NS (85) 0.479 *** (76) 0.219 NS (76) –0.702 *** (76) M a c a c a 79 0.626 *** (55) –0.283 * (55) –0.901 *** (55) –0.018 NS (55) –0.472 *** (55) 0.301 * (53) 0.111 NS (48) 0.329 * (48) –0.881 *** (48) M a c a c a 81 0.658 *** (46) 0.007 NS (46) –0.694 *** (46) –0.135 NS (46) –0.086 NS (66) 0.180 NS (63) 0.026 NS (17) 0.565 * (17) –0.748 *** (17) M a c a c a 83 0.705 *** (94) –0.006 NS (94) –0.649 *** (94) –0.150 NS (94) 0.032 NS (101) 0.543 *** (99) –0.119 NS (95) 0.744 *** (95) –0.675 *** (95) M a c a c a 84 0.722 *** (46) 0.163 NS (46) –0.497 *** (46) 0.033 NS (46) 0.084 NS (52) 0.434 ** (51) 0.628 *** (42) 0.544 *** (42) –0.292 NS (42) M a c a c a 791 0.440 * (22) 0.489 * (22) –0.461 * (22) 0.582 ** (22) –0.301 NS (40) 0.444 ** (40) Otole mu r 28 0.830 *** (21) 0.386 NS (21) –0.191 NS (21) 0.399 NS (21) –0.096 NS (21) 0.284 NS (21) 0.484 * (21) 0.924 *** (21) 0.119 NS (21) Otole mu r 63 0.916 *** (26) 0.411 * (26) 0.173 NS (26) 0.301 NS (26) –0.114 NS (26) 0.607 ** (26) 0.324 NS (24) 0.874 *** (24) –0.138 NS (24) Otole mu r 921 0.820 *** (89) –0.082 NS (89) –0.605 *** (89) –0.221 * (89) 0.280 ** (91) –0.150 NS (90) –0.224 * (87) 0.818 *** (87) –0.740 *** (87) Otole mu r 922 0.802 *** (13) 0.690 ** (13) 0.158 NS (13) 0.473 NS (13) 0.515 NS (13) 0.130 NS (13) –0.383 NS (9) 0.978 *** (9) –0.567 NS (9) Su s 101 0.917 *** (27) 0.571 ** (27) 0.271 NS (27) 0.535 ** (27) –0.035 NS (50) –0.103 NS (28) 0.658 *** (34) 0.873 *** (34) 0.285 NS (34) Su s 102 0.871 *** (45) 0.764 *** (45) 0.353 * (45) 0.681 *** (45) –0.179 NS (45) 0.402 ** (45) 0.388 ** (45) 0.938 *** (45) 0.134 NS (45) Su s 103 0.890 *** (49) –0.104 NS (49) –0.533 *** (49) –0.275 NS (49) 0.347 ** (66) –0.703 *** (53) 0.656 *** (33) 0.618 *** (33) 0.033 NS (33) Su s 104 0.827 *** (178) 0.027 NS (178) –0.489 *** (178) 0.053 NS (178) 0.082 NS (198) –0.111 NS (174) 0.122 NS (187) 0.849 *** (187) –0.363 *** (187) Su s 105 0.357 * (38) –0.163 NS (38) –0.907 *** (38) –0.138 NS (38) –0.020 NS (49) 0.186 NS (21) 0.309 * (49) 0.926 *** (49) –0.070 NS (49) Su s 106 0.118 NS (16) 0.165 NS (16) –0.926 *** (16) 0.203 NS (16) 0.053 NS (47) –0.237 NS (41) –0.028 NS (36) 0.640 *** (36) –0.721 *** (36) V a reci a 91 0.861 *** (51) –0.230 NS (51) –0.667 *** (51) –0.370 ** (51) 0.320 * (51) –0.413 ** (46) 0.248 NS (48) 0.888 *** (48) –0.188 NS (48) V a reci a 93 0.903 *** (44) –0.040 NS (44) –0.438 ** (44) –0.272 NS (44) 0.195 NS (51) –0.072 NS (27) –0.009 NS (28) 0.866 *** (28) –0.410 * (28) V a reci a 94 0.859 *** (52) –0.302 * (52) –0.658 *** (52) –0.270 NS (52) –0.161 NS (53) –0.016 NS (41) –0.189 NS (48) 0.943 *** (48) –0.481 *** (48) Val u
es are correlation coefficients (
P
earson’s
r
, or Spearman
’
s rho); sample size is
g
iven in parentheses. NS, not si
g nificant ( P >0.05); * P <0.05; ** P <0.01; *** P <0.001.
Sample sizes vary within an experiment beca
u
se not all variables co
u
ld be meas
u
Table · 3. Correl a tio n coefficie n ts for 2 d a t a Species Experiment Ma g nit u de v s rate Ma g nit u de v s load time Rate v s load time Ma g nit u de v s
load time STD
Ma g nit u de v s cycle time Ma g nit u de v s d u ty factor Ma g nit u de v s u nload time Ma g nit u de v s u nload rate Unload rate v s u nload time Aot u s 9 0.734 *** (251) –0.399 *** (251) 0.241 *** (251) –0.214 *** (251) –0.321 *** (384) –0.402 *** (167) –0.523 *** (195) 0.802 *** (195) 0.017 NS (195) Aot u s 10 0.612 *** (181) 0.058 NS (181) 0.797 *** (181) 0.013 NS (181) 0.099 NS (195) –0.235 ** (167) –0.458 *** (176) 0.524 *** (176) 0.425 *** (176) Aot u s 31 0.709 *** (112) –0.237 * (112) 0.394 *** (112) –0.049 NS (112) –0.108 NS (119) 0.088 NS (103) 0.021 NS (109) 0.825 *** (109) 0.535 *** (109) C a pr a 12 0.750 *** (20) 0.556 * (20) 0.928 *** (20) 0.553 * (20) 0.150 NS (24) 0.405 NS (20) –0.306 NS (24) 0.296 NS (24) 0.793 *** (24) C a pr a 14 0.900 *** (28) 0.648 *** (28) 0.894 *** (28) 0.666 *** (28) –0.141 NS (28) 0.624 *** (28) 0.061 NS (28) 0.846 *** (28) 0.556 ** (28) C a pr a 16 0.826 *** (16) 0.422 NS (16) 0.844 *** (16) 0.509 * (16) –0.516 * (22) 0.590 * (16) –0.307 NS (22) 0.942 *** (22) 0.023 NS (22) C a pr a 20 0.901 *** (48) 0.247 NS (48) 0.555 *** (48) 0.128 NS (44) 0.038 NS (44) –0.165 NS (44) –0.615 *** (48) 0.817 *** (48) –0.080 NS (48) C a pr a 33 0.461 * (28) –0.168 NS (28) 0.778 *** (28) –0.139 NS (28) 0.078 NS (41) –0.266 NS (26) –0.250 NS (26) 0.549 ** (26) 0.652 *** (26) C h loroceb u s 59 0.076 NS (25) –0.295 NS (25) 0.827 *** (25) –0.233 NS (25) –0.134 NS (63) –0.385 NS (14) –0.449 ** (34) 0.526 ** (34) 0.428 * (34) C h loroceb u s 96 0.350 ** (79) –0.257 * (79) 0.760 *** (79) –0.094 NS (79) –0.138 NS (87) –0.191 NS (75) –0.405 *** (82) 0.512 *** (82) 0.486 *** (82) Eq uu s 35 0.904 *** (39) 0.072 NS (39) 0.630 *** (39) 0.185 NS (35) –0.198 NS (36) –0.030 NS (33) –0.575 *** (38) 0.821 *** (38) –0.181 NS (38) Eq uu s 53 –0.104 NS (55) –0.459 *** (55) 0.956 *** (55) –0.516 *** (51) 0.437 *** (54) –0.524 *** (51) –0.006 NS (58) 0.481 *** (58) 0.860 *** (58) E u le mu r 67 0.811 *** (60) 0.233 NS (60) 0.688 *** (60) 0.164 NS (60) 0.282 ** (93) –0.016 NS (60) –0.313 ** (93) 0.659 *** (93) 0.414 *** (93) E u le mu r 70 0.826 *** (75) 0.404 *** (75) 0.814 *** (75) 0.274 * (75) –0.034 NS (83) 0.118 NS (75) –0.156 NS (80) 0.760 *** (80) 0.472 *** (80) E u le mu r 71 0.813 *** (232) 0.091 NS (232) 0.602 *** (232) 0.121 NS (232) 0.016 NS (235) 0.230 *** (230) 0.281 *** (231) 0.914 *** (231) 0.603 *** (231) L ama 40 0.303 NS (27) –0.087 NS (27) 0.849 *** (27) 0.025 NS (25) –0.391 NS (25) 0.248 NS (25) 0.275 NS (26) 0.482 * (26) 0.943 *** (26) L ama 50 0.696 *** (58) –0.213 NS (58) 0.489 *** (58) –0.229 NS (51) 0.204 NS (62) –0.537 *** (49) –0.399 ** (58) 0.714 *** (58) 0.384 ** (58) L ama 52 0.515 ** (32) –0.393 * (32) 0.560 *** (32) –0.287 NS (30) –0.157 NS (30) –0.159 NS (29) 0.109 NS (31) 0.537 ** (31) 0.846 *** (31) L ama 56 0.894 NS (4) 0.800 NS 6 8 4. 0 – ) 4( * 4 8 9. 0 ) 4( NS 6 3 5. 0 – ) 5 1( NS (7) –0.571 NS (7) 0.964 *** (7) M a c a c a 38 –0.199 * (162) –0.541 *** (162) 0.906 *** (162) –0.549 *** (162) 0.236 ** (167) –0.472 *** (158) –0.059 NS (162) 0.310 *** (162) 0.908 *** (162) M a c a c a 46 0.466 *** (58) 0.031 NS (58) 0.864 *** (58) –0.273 * (58) 0.002 NS (62) –0.231 NS (54) –0.070 NS (54) 0.602 *** (54) 0.703 *** (54) M a c a c a 48 0.665 *** (90) 0.100 NS (90) 0.789 *** (90) 0.079 NS (90) –0.007 NS (95) –0.013 NS (85) –0.290 ** (86) 0.503 *** (86) 0.702 *** (86) M a c a c a 79 0.541 *** (55) 0.033 NS (55) 0.810 *** (55) –0.157 NS (55) 0.450 *** (55) –0.281 * (53) –0.189 NS (53) 0.408 ** (53) 0.773 *** (53) M a c a c a 81 0.660 *** (65) 0.033 NS (65) 0.735 *** (65) 0.006 NS (65) 0.099 NS (66) –0.119 NS (63) –0.076 NS (64) 0.522 *** (64) 0.757 *** (64) M a c a c a 83 0.702 *** (100) –0.551 *** (100) 0.024 NS (100) –0.423 *** (100) –0.002 NS (101) –0.572 *** (99) –0.628 *** (100) 0.565 *** (100) 0.071 NS (100) M a c a c a 84 0.760 *** (52) –0.146 NS (52) 0.446 *** (52) –0.069 NS (52) –0.165 NS (52) –0.449 *** (51) –0.594 *** (51) 0.294 * (51) 0.510 *** (51) M a c a c a 791 0.653 *** (40) –0.130 NS (40) 0.580 *** (40) –0.295 NS (40) 0.225 NS (40) –0.248 NS (40) 0.220 NS (40) 0.732 *** (40) 0.788 *** (40) Otole mu r 28 0.775 *** (21) –0.321 NS (21) 0.345 NS (21) –0.087 NS (21) –0.128 NS (21) –0.097 NS (21) –0.821 *** (21) 0.865 *** (21) –0.486 * (21) Otole mu r 63 0.911 *** (26) –0.562 ** (26) –0.193 NS (26) –0.467 * (26) 0.072 NS (26) –0.599 ** (26) –0.557 ** (26) 0.863 *** (26) –0.083 NS (26) Otole mu r 921 0.834 *** (90) –0.175 NS (90) 0.403 *** (90) 0.005 NS (90) –0.266 * (91) 0.087 NS (90) –0.085 NS (91) 0.970 *** (91) 0.108 NS (91) Otole mu r 922 0.897 *** (13) –0.750 ** (13) –0.410 NS (13) –0.513 NS (13) –0.474 NS (13) –0.223 NS (13) –0.008 NS (13) 0.969 *** (13) 0.225 NS (13) Su s 101 0.826 *** (48) 0.069 NS (48) 0.566 *** (48) –0.091 NS (48) 0.320 * (50) –0.177 NS (28) 0.083 NS (29) 0.884 *** (29) 0.441 * (29) Su s 102 0.920 *** (45) –0.538 *** (45) –0.250 NS (45) –0.339 * (45) 0.138 NS (45) –0.369 * (45) –0.539 *** (45) 0.965 *** (45) –0.364 * (45) Su s 103 0.912 *** (65) 0.440 *** (65) 0.725 *** (65) 0.565 *** (65) –0.654 *** (66) 0.681 *** (53) 0.576 *** (53) 0.903 *** (53) 0.820 *** (53) Su s 104 0.900 *** (181) 0.084 NS (181) 0.453 *** (181) 0.115 NS (181) –0.189 ** (198) –0.099 NS (174) –0.383 *** (186) 0.799 *** (186) 0.179 * (186) Su s 105 –0.208 NS (22) –0.391 NS (22) 0.946 *** (22) –0.461 * (22) 0.230 NS (49) –0.601 ** (21) –0.320 NS (31) 0.912 *** (31) 0.093 NS (31) Su s 106 0.899 *** (42) 0.255 NS (42) 0.528 *** (42) –0.022 NS (42) 0.086 NS (47) –0.002 NS (41) 0.048 NS (46) 0.539 *** (46) 0.820 *** (46) V a reci a 91 0.857 *** (47) 0.128 NS (47) 0.586 *** (47) 0.226 NS (47) –0.252 NS (51) 0.371 * (46) –0.114 NS (48) 0.941 *** (48) 0.239 NS (48) V a reci a 93 0.779 *** (46) 0.153 NS (46) 0.265 NS (46) 0.254 NS (46) –0.177 NS (51) 0.115 NS (27) –0.367 * (29) 0.666 *** (29) 0.142 NS (29) V a reci a 94 0.892 *** (45) 0.444 ** (45) 0.784 *** (45) 0.269 NS (45) 0.209 NS (53) 0.092 NS (41) –0.241 NS (47) 0.951 *** (47) 0.101 NS (47) Val u
es are correlation coefficients (
P
earson
’
s
r
, or Spearman
’
s rho); sample size is
g
iven in parentheses. NS, not si
g nificant ( P >0.05); * P <0.05; ** P <0.01; *** P <0.001.
Sample sizes vary within an experiment beca
u
se not all variables co
u
ld be meas
u
significantly (P⭐0.001) correlated with unload time in 12 out of 40 experiments.
In the majority of experiments the highest correlation coefficients were observed between strain magnitude and load rate. Out of those experiments in which ⑀1 magnitude was significantly correlated with both load time and load rate, ⑀1 magnitude was only most highly correlated with load time in three experiments, one on goats and two on macaques. ⑀2 magnitude was not more highly correlated with load time than load rate in any experiment in which the correlation was significant.
In sum, principal strain magnitudes are most often and most highly correlated with estimates of loading rate rather than loading time. Principal strain magnitudes are also most often and most highly correlated with unloading rates, rather than unloading times.
Multiple regressions
Data on strain magnitude, strain rate and load time from 11 out of 40 experiments met the assumptions of multiple regression (i.e. were homoscedastic and normally distributed). A total of 16 multiple regression equations was calculated. In three experiments, both the ⑀1 and the ⑀2 data could be analyzed; in four experiments, only the ⑀1 data could be analyzed; and in
Fig.·4. Bivariate plots of ⑀1magnitude in microstrain (⑀) against
loading time (in s) (A) and loading rate in ⑀·s–1(B). Data recorded
during Experiment 9 on Aotuseating a range of foods. Note that there is not a significant correlation between strain magnitude and loading time (A), but there is a significant correlation between strain magnitude and loading rate (B). It is of interest that these patterns of relationship (or lack thereof) apply within most, but not all, of the different food types. Variations in these relationships within food type are not consistent across experiments.
Food Apple Apricot Carrot
Gummy bear Plantains Prune
0 2000 4000 6000 8000
100 200 300 400 500 ε1
magnitude (
με
)
ε1 load rate (με s–1)
0.05 0.10 0.15 0.20 100
200 300 400 500
ε1 load time (s)
B
A
Fig.·5. Bivariate plots of ⑀1 magnitude in microstrain (⑀) against
loading time (in s) (A) and loading rate in ⑀·s–1(B). Data recorded during Experiment 103 on Sus eating pig chow. The data are labeled by chewing sequence. Note that, across all chews, there is not a significant correlation between strain magnitude and loading time (A), but there is a significant correlation between strain magnitude and loading rate (B). Analyses of data within chewing sequences reveal that these patterns are also seen within sequences.
Sequence 1 2
1 2
0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
500 1000 1500
0 50
25 75 100 125 150
ε1
magnitude (
με
)
ε1 load rate (με s–1) ε1 load time (s)
B
Sequence
Food=pig chow
50
25 75 100 125 150
five experiments, only the ⑀2 data could be analyzed (Table·4, Table·5).
Multiple regressions of strain magnitude against load rate and load time are highly significant, with adjusted r2 values ranging from 0.669–0.996, all but one being >0.900. In all 16 cases, the beta (standardized partial slope) coefficients for load rate are higher (1.1–2.6⫻ higher) than those for load time. These results suggest that changes in load rate have a greater impact on strain magnitude than changes in loading time.
As expected, the magnitude of the correlation between load rate and load time is related to the ‘tolerance’ value. As the correlations between the independent variables (load rate and load time) increase, the tolerance values decrease, indicating effects of multicollinearity. Load rate and load time are significantly (P<0.05) correlated in 11 of the 16 calculations. In eight experiments, load rates decrease as load time increases
(negative correlations between ⑀1 rate and time; positive correlations between ⑀2 rate and time); in the other eight experiments, increases in load rate are associated with increases in load time. However, interaction effects between load rate and load time were not significant in any of the experiments, indicating that load rate and load time have independent effects on strain magnitude.
Regressions of strain magnitude against all variables together and separately (load rate, load time, load time STD, cycle time and duty factor) did not generate better models than regressions only on load rate and load time. When all variables were included, neither cycle time nor load time STD contributed significantly to the model. Similarly, when strain magnitude was regressed against load rate, load time and cycle time, only cycle time contributed significantly to the model in the case of Experiment 33 (⑀2) with an increase in adjusted r2
Fig.·6. Plots illustrating analysis of data from Experiment 16 on Capraeating hay. (A) Bivariate plot of ⑀2magnitude in microstrain (⑀) against
loading time (in s). (B) Bivariate plot of ⑀2magnitude in microstrain (⑀) against loading rate in ⑀·s–1. Regression equation of ⑀2magnitude
against loading rate in ⑀·s–1: ⑀2magnitude=–117.07+0.11⫻⑀2load rate. (C,D) Partial regression plots from multiple regression of ⑀2magnitude
in microstrain (⑀) against loading time (in s) and loading rate in ⑀·s–1. (E) Plot of residual ⑀
2magnitude (i.e. variance not explained by the
regression in B) against load time (s). (F) Bivariate plot of loading rate in ⑀·s–1, against loading time (in s). There is not a significant correlation between strain magnitude and loading time (A), but there is a significant correlation between strain magnitude and loading rate (B). Partial regression plots illustrate relationship between dependent variable (⑀2magnitude) and one independent variable, while holding the other variable
constant. These partial regression plots suggest close relationships between strain magnitude and each independent variable when controlling for the other because, as quantified here, strain magnitude must be nearly completely explained by a combination of load rate and load time. (F) Increases in loading rate are accompanied by increases in loading time, reinforcing the conclusion that load time and load rate are both strategies employed to increase strain magnitude. However, examination of bivariate plots A and B reveals that load time explains little of the variance in strain magnitude. Once the effect of strain rate is accounted for, there is a weak relationship between residual strain magnitude and load time, as illustrated in E, with load time explaining much less of the variance in strain magnitude than load rate. The data from this experiment consist of two separate chewing sequences. The data from the two sequences are indicated by separate symbols, showing that the effects revealed across the whole experiment also obtain within chewing sequences.
0.15 0.20 0.25 0.30
–2000 –1500 –1000 –400
–350 –300 –250 –200 –150
–750 –500 –250 0 250 500 –150
–100 –50 0 50 100
–0.06 –0.04 –0.02 0 0.02 0.04 0.06 –60
–40 –20 0 20 40 60
A
B
C
D
0.14 0.18 0.22 0.26 0.30
E
F
–2000 –1500 –1000
Sequence 1 2 –400
–350 –300 –250 –200 –150
Sequence 1 2
ε2
magnitude (
με
)
ε2 load rate (με s–1) ε2 load rate (με s–1)
ε2 load time (s) ε2 load time (s)
–60 –40 –20 0 20 40 60
ε2 load time (s)
Sequence 1 2
Residual
ε2
magnitude (
με
)
0.14 0.18 0.22 0.26 0.30
ε2 load time (s)
Sequence 1 2
ε2
load r
ate (
με
s
–1
from 0.906 to 0.948, and Experiment 20 (⑀1) (r2from 0.932 to 0.962). When strain magnitude was regressed against load rate, load time, and duty factor, only duty factor contributed significantly to the model in Experiment 20 (⑀1), and at P=0.042, and with only a slight increase in adjusted r2(from 0.932 to 0.959).
When cycle time and load time STD were combined with load rate and load time, cycle and load time STD only produced an increase in adjusted r2 in four experiments, and these increases were very modest (0.921 to 0.925; 0.984 to 0.986; 0.974 to 0.980; 0.962 to 0.981). Substituting cycle time and load time STD for load time produced comparably modest increases in adjusted r2in Experiment 16 (⑀2) (0.936 to 0.938) and (⑀1) (0.930 to 0.933), Experiment 33 (⑀2) (0.906 to 0.949), and Experiment 922 (⑀1) (0.963 to 0.964).
Overall, a simple regression model including only load rate and load time as independent variables best explains variance in strain magnitude. Cycle time, duty factor and load time STD
do not explain significant amounts of variance over and above that accounted for by load time itself.
Fig.·6 illustrates the relationships between ⑀2 magnitude and load time and load rate in the data from Experiment 16, as revealed by multiple regression analyses. The bivariate plots of ⑀2 magnitude against loading time and loading rate illustrate the lack of a significant correlation between strain magnitude and loading time (Fig.·6A), and the significant correlation between strain magnitude and loading rate (Fig.·6B). The partial regression plots illustrate the relationships between the dependent variable (⑀2 magnitude) and each independent variable while holding the other independent variable constant. These partial regression plots (Fig.·6C,D) reveal close relationships between strain magnitude and each independent variable when controlling for the other because, as quantified here, strain magnitude must be nearly completely explained by a combination of load rate and load time. However, the greater importance of loading rate in explaining variance in mandible
Table·4. Results of multiple regressionanalyses of 1magnitude (dependent variable) against load rate and load time
(independent variables)
Experiment Adjusted r2
Variable Beta
Zero-order correlations
Partial
correlations Tolerance
16 0.930 Load rate 1.222*** 0.906*** 0.963 0.536
Load time 0.465*** –0.368NS
0.804 0.536
Load rate vs load Time –0.681***
20 0.932 Load rate 1.042*** 0.745*** 0.965 0.812
Load time 0.684*** 0.232NS 0.924 0.812
Load rate vs load Time –0.434**
28 0.996 Load rate 0.938*** 0.916*** 0.998 0.964
Load time 0.565*** 0.411NS
0.994 0.964
Load rate vs load Time –0.173NS
53 0.974 Load rate 2.029*** 0.548*** 0.987 0.235
Load time 1.694*** –0.080NS
0.982 0.235
Load rate vs load Time –0.874***
56 0.962 Load rate 1.227*** 0.795*** 0.983 0.642
Load time 0.721*** –0.012NS
0.954 0.642
Load rate vs load Time 0.970***
102 0.996 Load rate 0.687*** 0.871*** 0.996 0.875
Load time 0.521*** 0.764*** 0.992 0.875
Load rate vsloadTime –0.353*
922 0.963 Load rate 0.711*** 0.802*** 0.976 0.975
· Load time 0.578*** 0.690** 0.956 0.975
load rate vsloadTime 0.158NS
Multiple regression statistics are presented by Experiment (identified in the 1st column). The adjusted r2 for the model is presented in the 2nd column, and the beta coefficients for strain magnitude against each individual variable (listed in 3rd column) are given with their significance levels (4th columns). The zero-order correlation coefficients from Table·2 are reproduced in the 5th column for ease of reference, along with their significance levels, and the partial correlation coefficients for the multiple regression model (6th column). The tolerance value is given in the last column. The tolerance for a variable is the proportion of the variance in that variable not accounted for by other independent variables in the model. A low value indicates that the variable contributes little to the model independent of the other variables, and is an indicator of multicollinearity.
strain magnitude is illustrated by Fig.·6E, a bivariate plot of the residual ⑀2 magnitude from Fig.·6B (i.e. the variance in ⑀2 magnitude not explained by the regression in B) against loading time. Once the effect of strain rate is accounted for, there is only a very weak relationship between residual strain magnitude and load time, as illustrated in Fig.·6E, with load time explaining much less of the variance in strain magnitude than load rate. Fig.·6F illustrates that increases in loading rate are also accompanied by increases in loading time, suggesting that increases in load magnitudes are accompanied by both increases in load rate and load time.
Discussion
This study uses mandibular corpus bone strain data to make inferences about how mammals modulate their bite forces during mastication. Two competing hypotheses were evaluated: that bite force is modulated primarily by modulating loading times, with consequent changes in chew cycle time and/or duty factor; or that bite force is modulated primarily by
modulating loading rates. Across all nine genera of mammals examined here, strain magnitude is predominantly correlated with strain load rate, and less often with cycle or loading time. In addition, strain magnitude is usually more highly correlated with strain loading rate than with loading time or cycle time. Multiple regression results confirm that changes in strain loading rate have a greater influence on strain magnitude than do changes in loading time, but they also suggest that when changes in strain loading rate are taken into account, changes in strain magnitude are consistently associated with increases in loading time. These results suggest that during rhythmic mastication in mammals, mandibular bone strain magnitude and, by inference, bite force, are increased primarily by increasing the rate at which the mandible is loaded and, to a lesser extent, by increases in the duration of loading.
Time-modulation of bite force
The results presented here stand in apparent contrast to reports by various workers that cycle duration increases with increasing hardness of foods during chewing, or of materials
Table·5. Multiple regression results 2magnitude (dependent variable) against load rate and load time (independent
variables)
Experiment Adjusted r2
Variable Beta
Zero-order correlations
Partial
correlations Tolerance
12 0.669 Load rate 1.688*** 0.750*** 0.756 0.138
Load time –1.011* 0.556* –0.376 0.138
Load rate vsloadTime 0.928***
16 0.936 Load rate 1.631*** 0.826*** 0.966 0.288
Load time –0.954*** 0.422NS
–0.909 0.288
Load rate vsloadTime 0.844***
28 0.994 Load rate 1.006*** 0.775*** 0.997 0.881
Load time –0.669*** –0.321NS
–0.993 0.881
Load rate vsloadTime 0.345NS
33 0.906 Load rate 1.496*** 0.461*** 0.954 0.395
Load time –1.331*** –0.168NS
–0.943 0.395
Load rate vsloadTime 0.778***
48 0.921 Load rate 1.557*** 0.665*** 0.960 0.377
Load time –1.129*** 0.060NS
–0.693 0.377
Load rate vsloadTime 0.789***
52 0.938 Load rate 1.071*** 0.515** 0.965 0.686
Load time –0.993*** –0.393* –0.960 0.686
Load rate vsloadTime 0.560***
63 0.984 Load rate 0.834*** 0.911*** 0.911 0.963
Load time –0.401*** –0.562** –0.562 0.963
Load rate vsloadTime –0.193NS
2 3 8 . 0 7
9 8 . 0
* * *
9 9 8 . 0
* * *
9 0 7 . 0 e
t a r d a o L 2
2 9
Load time –0.459*** 0.750** –0.750 0.832
Load rate vsloadTime –0.410NS
Multiple regression statistics are presented by Experiment (identified in the 1st column). The adjusted r2 for the model is presented in the 2nd column, and the beta coefficients for strain magnitude against each individual variable (listed in 3rd column) are given with their significance levels (4th columns). The zero-order correlation coefficients from Table 2 are reproduced in the 5th column for ease of reference, along with their significance levels, and the partial correlation coefficients for the multiple regression model (6th column). The tolerance value is given in the last column. The tolerance for a variable is the proportion of the variance in that variable not accounted for by other independent variables in the model. A low value indicates that the variable contributes little to the model independent of the other variables, and is an indicator of multicollinearity.
placed between the teeth during cortically evoked rhythmic jaw movements (CRJMs) (Hidaka et al., 1997; Lavigne et al., 1987; Liu et al., 1998; Liu et al., 1993; Plesh et al., 1986). If cycle duration increases with food hardness, why does mandibular loading time not reliably predict mandibular loading magnitude during rhythmic chewing? Variation in cycle duration need not necessarily translate into consistent variation in the duration of mandibular loading if changes in cycle duration are due to changes in opening and closing phase exclusive of the slow close/power stroke (SC/PS) phase (e.g. Plesh et al., 1986). Moreover, although it has also been reported that increases in food hardness can be associated with increases in the duration of SC (Thexton and Hiiemae, 1997; Yamada and Yamamura, 1996), SC duration need not be highly correlated with the durations of mandible loading and bite force generation (Hylander et al., 1987). In essence, it is not necessary that jaw kinematics and jaw kinetics be closely coupled.
One necessary result of this apparent decoupling of mandible loading duration and chew cycle duration is that there must be variation in chew duty factor. In this study, chew duty factor did show significant correlations with strain magnitude in approximately 25% of experiments, but it did not explain significantly more variance in multiple regression models, and was usually not significantly related to strain magnitude when other variables were included. If some animals do increase chew duty factors when increasing strain magnitudes and bite forces (Yamada and Yamamura, 1996), it is not a strategy adopted consistently within or across the mammalian species examined here (cf. Weijs and De Jongh, 1977).
Several studies have shown that increases in food ‘hardness’ are associated with increases in EMG burst duration (Hidaka et al., 1997; Lavigne et al., 1987; Liu et al., 1993; Lund et al., 1998; Morimoto et al., 1989). On the basis of these EMG data, it might seem reasonable to hypothesize that variation in bite force magnitudes in mammals is significantly correlated with variation in the duration of force generation. However, in those studies in which test strips of increasing hardness or steel balls were introduced between the teeth during CRJMs, increases in chew cycle time and durations of muscle activity were also accompanied by increases in muscle activity amplitudes (Hidaka et al., 1997; Lavigne et al., 1987; Liu et al., 1993; Lund et al., 1998; Morimoto et al., 1989). Hidaka et al. also found that increases in bite force are not accompanied by increases in bite load duration, but only in load rate. Harder foods did not elicit significant increases in masseter burst duration [fig.·11 in Liu et al. (Liu et al., 1993)] although there were increases in burst amplitude, suggesting again that increases in rate of force development must be more important for increasing bite force magnitudes than duration of force development.
In sum, the data suggesting that increases in bite force during CRJMs are associated with increases in jaw closer muscle burst duration are quite limited, and are not obviously related to patterns of bite force modulation. Perhaps the most important question about these studies is whether force is modulated the same way during CRJMs in anesthetized animals as in mastication in awake, alert animals. These considerations
require that we turn our attention to data from chewing in awake, alert animals.
Data from rabbits during rhythmic chewing show that burst durations in superficial masseter (measured from onset to offset) are longer during chewing of raw rice, of intermediate duration during pellet chewing, and shortest when chewing bread (Kakizaki et al., 2002; Weijs and Dantuma, 1981; Yamada and Haraguchi, 1995; Yamada and Yamamura, 1996). In humans, increases in gum hardness are associated with increases in masseter muscle burst duration measured from onset to offset (Plesh et al., 1986). [Other studies reporting changes in EMG activity with food hardness integrate the activity over various time periods, so the effect of hardness on burst durations cannot be definitively assessed (Agrawal et al., 1998; Foster et al., 2006; Lassauzay et al., 2000; Peyron et al., 2002; Plesh et al., 1986).] In cats, foods more resistant to compression tests (raw beef versus cooked chicken) elicit higher EMG burst durations, amplitude and spikes per unit time in some adductor muscles and not others (Gorniak and Gans, 1980).
Although all of these studies report increases in overall EMG burst durations in association with increases in food hardness, they also report increases in burst amplitude. However, it is not clear that overall EMG burst durations are indicative of how muscle force is modulated during the loading phase, nor how this relates to bite force generation. Without explicitly considering the relationship between changing food hardness and the rate of muscle force recruitment during loading (estimated as amplitude/time from onset to peak), data on overall muscle burst duration do not contradict the hypothesized importance of rate-modulation of muscle and bite force.
Rate-modulation of bite force